Table of Contents
Which differential equation has no solution?
In the mathematical study of partial differential equations, Lewy’s example is a celebrated example, due to Hans Lewy, of a linear partial differential equation with no solutions. It shows that the analog of the Cauchy–Kovalevskaya theorem does not hold in the smooth category.
Do differential equations always have a solution?
Not all differential equations will have solutions so it’s useful to know ahead of time if there is a solution or not. This question is usually called the existence question in a differential equations course.
What does it mean to find the particular solution to a differential equation?
A Particular Solution of a differential equation is a solution obtained from the General Solution by assigning specific values to the arbitrary constants.
How do we identify the order of a differential equation?
The order of a differential equation is determined by the highest-order derivative; the degree is determined by the highest power on a variable. The higher the order of the differential equation, the more arbitrary constants need to be added to the general solution.
What does the uniqueness of solutions of a differential equation Mean?
Uniqueness of solutions tells us that the integral curves for a differential equation cannot cross. x ( t) = x 0 + ∫ t 0 t f ( s, x ( s)) d s.
Do solutions to first order differential equations exist and are unique?
The existence and uniqueness of solutions will prove to be very important—even when we consider applications of differential equations. The following theorem tells us that solutions to first-order differential equations exist and are unique under certain reasonable conditions. Theorem 1.6.1. Existence and Uniqueness Theorem.
Why do we say “a particular solution” instead of the particular solution?
Notice in the last example that we kept saying “a” particular solution, not “the” particular solution. This is because there are other possibilities out there for the particular solution we’ve just managed to find one of them. Any of them will work when it comes to writing down the general solution to the differential equation. Speaking of which…
How do you find the coefficient of a differential equation?
The method is quite simple. All that we need to do is look at g(t) g ( t) and make a guess as to the form of Y P (t) Y P ( t) leaving the coefficient (s) undetermined (and hence the name of the method). Plug the guess into the differential equation and see if we can determine values of the coefficients.